Question

Two parallel chords of a circle of radius $2$ are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the centre, angles of $\frac{\pi }{k}$ and $\frac{2\pi }{k}$, where $k>0$, then the value of $[k]$ is ______
[Note : $[k]$denotes the largest integer less than or equal to $k$ ].

Answer 1

Answer: 2

$2\cos \frac{\pi }{2k}+2\cos \frac{\pi }{k}=\sqrt{3}+1$
$\cos \frac{\pi }{2k}+\cos \frac{\pi }{k}=\frac{\sqrt{3}+1}{2}$
Let $\frac{\pi }{k}=0,\cos \theta +\cos \frac{\theta }{2}=\frac{\sqrt{3}+1}{2}$
$\Rightarrow 2{\cos }^2\frac{\theta }{2}-1+\cos \frac{\theta }{2}=\frac{\sqrt{3}+1}{2}$
$\cos \frac{\theta }{2}=t$
$2t^2+t-\frac{\sqrt{3}+3}{2}=0$
$t=\frac{-1\pm \sqrt{1+4(3+\sqrt{3})}}{4}$
$=\frac{-1\pm (2\sqrt{3}+1)}{4}=\frac{-2-2\sqrt{3}}{4},\frac{\sqrt{3}}{2}$
$\because t\in [-1,1],\cos \frac{\theta }{2}=\frac{\sqrt{3}}{2}$
$\frac{\theta }{2}=\frac{\pi }{6}$
$\Rightarrow k=3$